20.4: Homeostatic Regulation of the Vascular System
Learning Objectives
- Discuss the mechanisms involved in the neural regulation of vascular homeostasis
- Describe the contribution of a variety of hormones to the renal regulation of blood pressure
- Identify the effects of exercise on vascular homeostasis
- Discuss how hypertension, hemorrhage, and circulatory shock affect vascular health
In order to maintain homeostasis in the cardiovascular system and provide adequate blood to the tissues, blood flow must be redirected continually to the tissues as they become more active. In a very real sense, the cardiovascular system engages in resource allocation, because there is not enough blood flow to distribute blood equally to all tissues simultaneously. For example, when an individual is exercising, more blood will be directed to skeletal muscles, the heart, and the lungs. Following a meal, more blood is directed to the digestive system.
Table \(\PageIndex{1}\) provides the distribution of systemic blood at rest and during exercise. Although most of the data appears logical, the values for the distribution of blood to the integument may seem surprising. During exercise, the body distributes more blood to the body surface where it can dissipate the excess heat generated by increased activity into the environment.
| Organ |
Resting (mL/min) |
Mild exercise (mL/min) |
Maximal exercise (mL/min) |
|---|---|---|---|
| Skeletal muscle | 1200 | 4500 | 12,500 |
| Heart | 250 | 350 | 750 |
| Brain | 750 | 750 | 750 |
| Integument | 500 | 1500 | 1900 |
| Kidney | 1100 | 900 | 600 |
| Gastrointestinal | 1400 | 1100 | 600 |
|
Others (i.e., liver, spleen) |
600 | 400 | 400 |
| Total | 5800 | 9500 | 17,500 |
Three homeostatic mechanisms ensure adequate blood flow, blood pressure, distribution, and ultimately perfusion: neural, endocrine, and autoregulatory mechanisms. They are summarized in Figure \(\PageIndex{1}\).
Neural Regulation
The nervous system plays a critical role in the regulation of vascular homeostasis. The primary regulatory sites include the cardiovascular centers in the brain that control both cardiac and vascular functions. In addition, more generalized neural responses from the limbic system and the autonomic nervous system are factors.
The Cardiovascular Centers in the Brain
Neurological regulation of blood pressure and flow depends on the cardiovascular centers located in the medulla oblongata. This cluster of neurons responds to changes in blood pressure as well as blood concentrations of oxygen, carbon dioxide, and hydrogen ions. The cardiovascular center contains three distinct paired components:
- The cardioaccelerator centers stimulate cardiac function by regulating heart rate and stroke volume via sympathetic stimulation from the cardiac accelerator nerve.
- The cardioinhibitor centers slow cardiac function by decreasing heart rate and stroke volume via parasympathetic stimulation from the vagus nerve.
- The vasomotor centers control vessel tone or contraction of the smooth muscle in the tunica media. Changes in diameter affect peripheral resistance, pressure, and flow, which affect cardiac output. The majority of these neurons act via the release of the neurotransmitter norepinephrine from sympathetic neurons.
Although each center functions independently, they are not anatomically distinct.
There is also a small population of neurons that control vasodilation in the vessels of the brain and skeletal muscles by relaxing the smooth muscle fibers in the vessel tunics. Many of these are cholinergic neurons, that is, they release acetylcholine, which in turn stimulates the vessels’ endothelial cells to release nitric oxide (NO), which causes vasodilation. Others release norepinephrine that binds to β 2 receptors. A few neurons release NO directly as a neurotransmitter.
Recall that mild stimulation of the skeletal muscles maintains muscle tone. A similar phenomenon occurs with vascular tone in vessels. As noted earlier, arterioles are normally partially constricted: With maximal stimulation, their radius may be reduced to one-half of the resting state. Full dilation of most arterioles requires that this sympathetic stimulation be suppressed. When it is, an arteriole can expand by as much as 150 percent. Such a significant increase can dramatically affect resistance, pressure, and flow.
Baroreceptor Reflexes
Baroreceptors are specialized stretch receptors located within thin areas of blood vessels and heart chambers that respond to the degree of stretch caused by the presence of blood. They send impulses to the cardiovascular center to regulate blood pressure. Vascular baroreceptors are found primarily in sinuses (small cavities) within the aorta and carotid arteries: The aortic sinuses are found in the walls of the ascending aorta just superior to the aortic valve, whereas the carotid sinuses are in the base of the internal carotid arteries. There are also low-pressure baroreceptors located in the walls of the venae cavae and right atrium.
When blood pressure increases, the baroreceptors are stretched more tightly and initiate action potentials at a higher rate. At lower blood pressures, the degree of stretch is lower and the rate of firing is slower. When the cardiovascular center in the medulla oblongata receives this input, it triggers a reflex that maintains homeostasis (Figure \(\PageIndex{2}\)):
- When blood pressure rises too high, the baroreceptors fire at a higher rate and trigger parasympathetic stimulation of the heart. As a result, cardiac output falls. Sympathetic stimulation of the peripheral arterioles will also decrease, resulting in vasodilation. Combined, these activities cause blood pressure to fall.
- When blood pressure drops too low, the rate of baroreceptor firing decreases. This will trigger an increase in sympathetic stimulation of the heart, causing cardiac output to increase. It will also trigger sympathetic stimulation of the peripheral vessels, resulting in vasoconstriction. Combined, these activities cause blood pressure to rise.
The baroreceptors in the venae cavae and right atrium monitor blood pressure as the blood returns to the heart from the systemic circulation. Normally, blood flow into the aorta is the same as blood flow back into the right atrium. If blood is returning to the right atrium more rapidly than it is being ejected from the left ventricle, the atrial receptors will stimulate the cardiovascular centers to increase sympathetic firing and increase cardiac output until homeostasis is achieved. The opposite is also true. This mechanism is referred to as the atrial reflex .
Chemoreceptor Reflexes
In addition to the baroreceptors are chemoreceptors that monitor levels of oxygen, carbon dioxide, and hydrogen ions (pH), and thereby contribute to vascular homeostasis. Chemoreceptors monitoring the blood are located in close proximity to the baroreceptors in the aortic and carotid sinuses. They signal the cardiovascular center as well as the respiratory centers in the medulla oblongata.
Since tissues consume oxygen and produce carbon dioxide and acids as waste products, when the body is more active, oxygen levels fall and carbon dioxide levels rise as cells undergo cellular respiration to meet the energy needs of activities. This causes more hydrogen ions to be produced, causing the blood pH to drop. When the body is resting, oxygen levels are higher, carbon dioxide levels are lower, more hydrogen is bound, and pH rises. (Seek additional content for more detail about pH.)
The chemoreceptors respond to increasing carbon dioxide and hydrogen ion levels (falling pH) by stimulating the cardioaccelerator and vasomotor centers, increasing cardiac output and constricting peripheral vessels. The cardioinhibitor centers are suppressed. With falling carbon dioxide and hydrogen ion levels (increasing pH), the cardioinhibitor centers are stimulated, and the cardioaccelerator and vasomotor centers are suppressed, decreasing cardiac output and causing peripheral vasodilation. In order to maintain adequate supplies of oxygen to the cells and remove waste products such as carbon dioxide, it is essential that the respiratory system respond to changing metabolic demands. In turn, the cardiovascular system will transport these gases to the lungs for exchange, again in accordance with metabolic demands. This interrelationship of cardiovascular and respiratory control cannot be overemphasized.
Other neural mechanisms can also have a significant impact on cardiovascular function. These include the limbic system that links physiological responses to psychological stimuli, as well as generalized sympathetic and parasympathetic stimulation.
Endocrine Regulation
Endocrine control over the cardiovascular system involves the catecholamines, epinephrine and norepinephrine, as well as several hormones that interact with the kidneys in the regulation of blood volume.
Epinephrine and Norepinephrine
The catecholamines epinephrine and norepinephrine are released by the adrenal medulla, and enhance and extend the body’s sympathetic or “fight-or-flight” response (see Figure \(\PageIndex{1}\)). They increase heart rate and force of contraction, while temporarily constricting blood vessels to organs not essential for flight-or-fight responses and redirecting blood flow to the liver, muscles, and heart.
Antidiuretic Hormone
Antidiuretic hormone (ADH), also known as vasopressin, is secreted by the cells in the hypothalamus and transported via the hypothalamic-hypophyseal tracts to the posterior pituitary where it is stored until released upon nervous stimulation. The primary trigger prompting the hypothalamus to release ADH is increasing osmolarity of tissue fluid, usually in response to significant loss of blood volume. ADH signals its target cells in the kidneys to reabsorb more water, thus preventing the loss of additional fluid in the urine. This will increase overall fluid levels and help restore blood volume and pressure. In addition, ADH constricts peripheral vessels.
Renin-Angiotensin-Aldosterone Mechanism
The renin-angiotensin-aldosterone mechanism has a major effect upon the cardiovascular system (Figure \(\PageIndex{3}\)). Renin is an enzyme, although because of its importance in the renin-angiotensin-aldosterone pathway, some sources identify it as a hormone. Specialized cells in the kidneys found in the juxtaglomerular apparatus respond to decreased blood flow by secreting renin into the blood. Renin converts the plasma protein angiotensinogen, which is produced by the liver, into its active form—angiotensin I. Angiotensin I circulates in the blood and is then converted into angiotensin II in the lungs. This reaction is catalyzed by the enzyme angiotensin-converting enzyme (ACE).
Angiotensin II is a powerful vasoconstrictor, greatly increasing blood pressure. It also stimulates the release of ADH and aldosterone, a hormone produced by the adrenal cortex. Aldosterone increases the reabsorption of sodium into the blood by the kidneys. Since water follows sodium, this increases the reabsorption of water. This in turn increases blood volume, raising blood pressure. Angiotensin II also stimulates the thirst center in the hypothalamus, so an individual will likely consume more fluids, again increasing blood volume and pressure.
Erythropoietin
Erythropoietin (EPO) is released by the kidneys when blood flow and/or oxygen levels decrease. EPO stimulates the production of erythrocytes within the bone marrow. Erythrocytes are the major formed element of the blood and may contribute 40 percent or more to blood volume, a significant factor of viscosity, resistance, pressure, and flow. In addition, EPO is a vasoconstrictor. Overproduction of EPO or excessive intake of synthetic EPO, often to enhance athletic performance, will increase viscosity, resistance, and pressure, and decrease flow in addition to its contribution as a vasoconstrictor.
Atrial Natriuretic Hormone
Secreted by cells in the atria of the heart, atrial natriuretic hormone (ANH) (also known as atrial natriuretic peptide) is secreted when blood volume is high enough to cause extreme stretching of the cardiac cells. Cells in the ventricle produce a hormone with similar effects, called B-type natriuretic hormone. Natriuretic hormones are antagonists to angiotensin II. They promote loss of sodium and water from the kidneys, and suppress renin, aldosterone, and ADH production and release. All of these actions promote loss of fluid from the body, so blood volume and blood pressure drop.
Autoregulation of Perfusion
As the name would suggest, autoregulation mechanisms require neither specialized nervous stimulation nor endocrine control. Rather, these are local, self-regulatory mechanisms that allow each region of tissue to adjust its blood flow—and thus its perfusion. These local mechanisms include chemical signals and myogenic controls.
Chemical Signals Involved in Autoregulation
Chemical signals work at the level of the precapillary sphincters to trigger either constriction or relaxation. As you know, opening a precapillary sphincter allows blood to flow into that particular capillary, whereas constricting a precapillary sphincter temporarily shuts off blood flow to that region. The factors involved in regulating the precapillary sphincters include the following:
- Opening of the sphincter is triggered in response to decreased oxygen concentrations; increased carbon dioxide concentrations; increasing levels of lactic acid or other byproducts of cellular metabolism; increasing concentrations of potassium ions or hydrogen ions (falling pH); inflammatory chemicals such as histamines; and increased body temperature. These conditions in turn stimulate the release of NO, a powerful vasodilator, from endothelial cells (see Figure \(\PageIndex{1}\)).
- Contraction of the precapillary sphincter is triggered by the opposite levels of the regulators, which prompt the release of endothelins, powerful vasoconstricting peptides secreted by endothelial cells. Platelet secretions and certain prostaglandins may also trigger constriction.
Again, these factors alter tissue perfusion via their effects on the precapillary sphincter mechanism, which regulates blood flow to capillaries. Since the amount of blood is limited, not all capillaries can fill at once, so blood flow is allocated based upon the needs and metabolic state of the tissues as reflected in these parameters. Bear in mind, however, that dilation and constriction of the arterioles feeding the capillary beds is the primary control mechanism.
The Myogenic Response
The myogenic response is a reaction to the stretching of the smooth muscle in the walls of arterioles as changes in blood flow occur through the vessel. This may be viewed as a largely protective function against dramatic fluctuations in blood pressure and blood flow to maintain homeostasis. If perfusion of an organ is too low (ischemia), the tissue will experience low levels of oxygen (hypoxia). In contrast, excessive perfusion could damage the organ’s smaller and more fragile vessels. The myogenic response is a localized process that serves to stabilize blood flow in the capillary network that follows that arteriole.
When blood flow is low, the vessel’s smooth muscle will be only minimally stretched. In response, it relaxes, allowing the vessel to dilate and thereby increase the movement of blood into the tissue. When blood flow is too high, the smooth muscle will contract in response to the increased stretch, prompting vasoconstriction that reduces blood flow.
Figure \(\PageIndex{4}\) summarizes the effects of nervous, endocrine, and local controls on arterioles.
Effect of Exercise on Vascular Homeostasis
The heart is a muscle and, like any muscle, it responds dramatically to exercise. For a healthy young adult, cardiac output (heart rate × stroke volume) increases in the nonathlete from approximately 5.0 liters (5.25 quarts) per minute to a maximum of about 20 liters (21 quarts) per minute. Accompanying this will be an increase in blood pressure from about 120/80 to 185/75. However, well-trained aerobic athletes can increase these values substantially. For these individuals, cardiac output soars from approximately 5.3 liters (5.57 quarts) per minute resting to more than 30 liters (31.5 quarts) per minute during maximal exercise. Along with this increase in cardiac output, blood pressure increases from 120/80 at rest to 200/90 at maximum values.
In addition to improved cardiac function, exercise increases the size and mass of the heart. The average weight of the heart for the nonathlete is about 300 g, whereas in an athlete it will increase to 500 g. This increase in size generally makes the heart stronger and more efficient at pumping blood, increasing both stroke volume and cardiac output.
Tissue perfusion also increases as the body transitions from a resting state to light exercise and eventually to heavy exercise (see Figure \(\PageIndex{4}\)). These changes result in selective vasodilation in the skeletal muscles, heart, lungs, liver, and integument. Simultaneously, vasoconstriction occurs in the vessels leading to the kidneys and most of the digestive and reproductive organs. The flow of blood to the brain remains largely unchanged whether at rest or exercising, since the vessels in the brain largely do not respond to regulatory stimuli, in most cases, because they lack the appropriate receptors.
As vasodilation occurs in selected vessels, resistance drops and more blood rushes into the organs they supply. This blood eventually returns to the venous system. Venous return is further enhanced by both the skeletal muscle and respiratory pumps. As blood returns to the heart more quickly, preload rises and the Frank-Starling principle tells us that contraction of the cardiac muscle in the atria and ventricles will be more forceful. Eventually, even the best-trained athletes will fatigue and must undergo a period of rest following exercise. Cardiac output and distribution of blood then return to normal.
Regular exercise promotes cardiovascular health in a variety of ways. Because an athlete’s heart is larger than a nonathlete’s, stroke volume increases, so the athletic heart can deliver the same amount of blood as the nonathletic heart but with a lower heart rate. This increased efficiency allows the athlete to exercise for longer periods of time before muscles fatigue and places less stress on the heart. Exercise also lowers overall cholesterol levels by removing from the circulation a complex form of cholesterol, triglycerides, and proteins known as low-density lipoproteins (LDLs), which are widely associated with increased risk of cardiovascular disease. Although there is no way to remove deposits of plaque from the walls of arteries other than specialized surgery, exercise does promote the health of vessels by decreasing the rate of plaque formation and reducing blood pressure, so the heart does not have to generate as much force to overcome resistance.
Generally as little as 30 minutes of noncontinuous exercise over the course of each day has beneficial effects and has been shown to lower the rate of heart attack by nearly 50 percent. While it is always advisable to follow a healthy diet, stop smoking, and lose weight, studies have clearly shown that fit, overweight people may actually be healthier overall than sedentary slender people. Thus, the benefits of moderate exercise are undeniable.
Clinical Considerations in Vascular Homeostasis
Any disorder that affects blood volume, vascular tone, or any other aspect of vascular functioning is likely to affect vascular homeostasis as well. That includes hypertension, hemorrhage, and shock.
Hypertension and Hypotension
Chronically elevated blood pressure is known clinically as hypertension . It is defined as chronic and persistent blood pressure measurements of 140/90 mm Hg or above. Pressures between 120/80 and 140/90 mm Hg are defined as prehypertension. About 68 million Americans currently suffer from hypertension. Unfortunately, hypertension is typically a silent disorder; therefore, hypertensive patients may fail to recognize the seriousness of their condition and fail to follow their treatment plan. The result is often a heart attack or stroke. Hypertension may also lead to an aneurism (ballooning of a blood vessel caused by a weakening of the wall), peripheral arterial disease (obstruction of vessels in peripheral regions of the body), chronic kidney disease, or heart failure.
Hemorrhage
Minor blood loss is managed by hemostasis and repair. Hemorrhage is a loss of blood that cannot be controlled by hemostatic mechanisms. Initially, the body responds to hemorrhage by initiating mechanisms aimed at increasing blood pressure and maintaining blood flow. Ultimately, however, blood volume will need to be restored, either through physiological processes or through medical intervention.
In response to blood loss, stimuli from the baroreceptors trigger the cardiovascular centers to stimulate sympathetic responses to increase cardiac output and vasoconstriction. This typically prompts the heart rate to increase to about 180–200 contractions per minute, restoring cardiac output to normal levels. Vasoconstriction of the arterioles increases vascular resistance, whereas constriction of the veins increases venous return to the heart. Both of these steps will help increase blood pressure. Sympathetic stimulation also triggers the release of epinephrine and norepinephrine, which enhance both cardiac output and vasoconstriction. If blood loss were less than 20 percent of total blood volume, these responses together would usually return blood pressure to normal and redirect the remaining blood to the tissues.
Additional endocrine involvement is necessary, however, to restore the lost blood volume. The angiotensin-renin-aldosterone mechanism stimulates the thirst center in the hypothalamus, which increases fluid consumption to help restore the lost blood. More importantly, it increases renal reabsorption of sodium and water, reducing water loss in urine output. The kidneys also increase the production of EPO, stimulating the formation of erythrocytes that not only deliver oxygen to the tissues but also increase overall blood volume. Figure \(\PageIndex{5}\) summarizes the responses to loss of blood volume.
Circulatory Shock
The loss of too much blood may lead to circulatory shock , a life-threatening condition in which the circulatory system is unable to maintain blood flow to adequately supply sufficient oxygen and other nutrients to the tissues to maintain cellular metabolism. It should not be confused with emotional or psychological shock. Typically, the patient in circulatory shock will demonstrate an increased heart rate but decreased blood pressure, but there are cases in which blood pressure will remain normal. Urine output will fall dramatically, and the patient may appear confused or lose consciousness. Urine output less than 1 mL/kg body weight/hour is cause for concern. Unfortunately, shock is an example of a positive-feedback loop that, if uncorrected, may lead to the death of the patient.
There are several recognized forms of shock:
- Hypovolemic shock in adults is typically caused by hemorrhage, although in children it may be caused by fluid losses related to severe vomiting or diarrhea. Other causes for hypovolemic shock include extensive burns, exposure to some toxins, and excessive urine loss related to diabetes insipidus or ketoacidosis. Typically, patients present with a rapid, almost tachycardic heart rate; a weak pulse often described as “thread;” cool, clammy skin, particularly in the extremities, due to restricted peripheral blood flow; rapid, shallow breathing; hypothermia; thirst; and dry mouth. Treatments generally involve providing intravenous fluids to restore the patient to normal function and various drugs such as dopamine, epinephrine, and norepinephrine to raise blood pressure.
- Cardiogenic shock results from the inability of the heart to maintain cardiac output. Most often, it results from a myocardial infarction (heart attack), but it may also be caused by arrhythmias, valve disorders, cardiomyopathies, cardiac failure, or simply insufficient flow of blood through the cardiac vessels. Treatment involves repairing the damage to the heart or its vessels to resolve the underlying cause, rather than treating cardiogenic shock directly.
- Vascular shock occurs when arterioles lose their normal muscular tone and dilate dramatically. It may arise from a variety of causes, and treatments almost always involve fluid replacement and medications, called inotropic or pressor agents, which restore tone to the muscles of the vessels. In addition, eliminating or at least alleviating the underlying cause of the condition is required. This might include antibiotics and antihistamines, or select steroids, which may aid in the repair of nerve damage. A common cause is sepsis (or septicemia), also called “blood poisoning,” which is a widespread bacterial infection that results in an organismal-level inflammatory response known as septic shock . Neurogenic shock is a form of vascular shock that occurs with cranial or spinal injuries that damage the cardiovascular centers in the medulla oblongata or the nervous fibers originating from this region. Anaphylactic shock is a severe allergic response that causes the widespread release of histamines, triggering vasodilation throughout the body.
- Obstructive shock , as the name would suggest, occurs when a significant portion of the vascular system is blocked. It is not always recognized as a distinct condition and may be grouped with cardiogenic shock, including pulmonary embolism and cardiac tamponade. Treatments depend upon the underlying cause and, in addition to administering fluids intravenously, often include the administration of anticoagulants, removal of fluid from the pericardial cavity, or air from the thoracic cavity, and surgery as required. The most common cause is a pulmonary embolism, a clot that lodges in the pulmonary vessels and interrupts blood flow. Other causes include stenosis of the aortic valve; cardiac tamponade, in which excess fluid in the pericardial cavity interferes with the ability of the heart to fully relax and fill with blood (resulting in decreased preload); and a pneumothorax, in which an excessive amount of air is present in the thoracic cavity, outside of the lungs, which interferes with venous return, pulmonary function, and delivery of oxygen to the tissues.
Chapter Review
Neural, endocrine, and autoregulatory mechanisms affect blood flow, blood pressure, and eventually perfusion of blood to body tissues. Neural mechanisms include the cardiovascular centers in the medulla oblongata, baroreceptors in the aorta and carotid arteries and right atrium, and associated chemoreceptors that monitor blood levels of oxygen, carbon dioxide, and hydrogen ions. Endocrine controls include epinephrine and norepinephrine, as well as ADH, the renin-angiotensin-aldosterone mechanism, ANH, and EPO. Autoregulation is the local control of vasodilation and constriction by chemical signals and the myogenic response. Exercise greatly improves cardiovascular function and reduces the risk of cardiovascular diseases, including hypertension, a leading cause of heart attacks and strokes. Significant hemorrhage can lead to a form of circulatory shock known as hypovolemic shock. Sepsis, obstruction, and widespread inflammation can also cause circulatory shock.
Interactive Link Questions
Listen to this CDC podcast to learn about hypertension, often described as a “silent killer.” What steps can you take to reduce your risk of a heart attack or stroke?
Answer: Take medications as prescribed, eat a healthy diet, exercise, and don’t smoke.
Review Questions
Q. Clusters of neurons in the medulla oblongata that regulate blood pressure are known collectively as ________.
A. baroreceptors
B. angioreceptors
C. the cardiomotor mechanism
D. the cardiovascular center
Answer: D
Q. In the renin-angiotensin-aldosterone mechanism, ________.
A. decreased blood pressure prompts the release of renin from the liver
B. aldosterone prompts increased urine output
C. aldosterone prompts the kidneys to reabsorb sodium
D. all of the above
Answer: C
Q. In the myogenic response, ________.
A. muscle contraction promotes venous return to the heart
B. ventricular contraction strength is decreased
C. vascular smooth muscle responds to stretch
D. endothelins dilate muscular arteries
Answer: C
Q. A form of circulatory shock common in young children with severe diarrhea or vomiting is ________.
A. hypovolemic shock
B. anaphylactic shock
C. obstructive shock
D. hemorrhagic shock
Answer: A
Critical Thinking Questions
Q. A patient arrives in the emergency department with a blood pressure of 70/45 confused and complaining of thirst. Why?
A. This blood pressure is insufficient to circulate blood throughout the patient’s body and maintain adequate perfusion of the patient’s tissues. Ischemia would prompt hypoxia, including to the brain, prompting confusion. The low blood pressure would also trigger the renin-angiotensin-aldosterone mechanism, and release of aldosterone would stimulate the thirst mechanism in the hypothalamus.
Q. Nitric oxide is broken down very quickly after its release. Why?
A. Nitric oxide is a very powerful local vasodilator that is important in the autoregulation of tissue perfusion. If it were not broken down very quickly after its release, blood flow to the region could exceed metabolic needs.
References
Centers for Disease Control and Prevention (US). Getting blood pressure under control: high blood pressure is out of control for too many Americans [Internet]. Atlanta (GA); [cited 2013 Apr 26]. Available from: www.cdc.gov/features/vitalsigns/hypertension/
Glossary
- anaphylactic shock
- type of shock that follows a severe allergic reaction and results from massive vasodilation
- aortic sinuses
- small pockets in the ascending aorta near the aortic valve that are the locations of the baroreceptors (stretch receptors) and chemoreceptors that trigger a reflex that aids in the regulation of vascular homeostasis
- atrial reflex
- mechanism for maintaining vascular homeostasis involving atrial baroreceptors: if blood is returning to the right atrium more rapidly than it is being ejected from the left ventricle, the atrial receptors will stimulate the cardiovascular centers to increase sympathetic firing and increase cardiac output until the situation is reversed; the opposite is also true
- cardiogenic shock
- type of shock that results from the inability of the heart to maintain cardiac output
- carotid sinuses
- small pockets near the base of the internal carotid arteries that are the locations of the baroreceptors and chemoreceptors that trigger a reflex that aids in the regulation of vascular homeostasis
- circulatory shock
- also simply called shock; a life-threatening medical condition in which the circulatory system is unable to supply enough blood flow to provide adequate oxygen and other nutrients to the tissues to maintain cellular metabolism
- hypertension
- chronic and persistent blood pressure measurements of 140/90 mm Hg or above
- hypovolemic shock
- type of circulatory shock caused by excessive loss of blood volume due to hemorrhage or possibly dehydration
- myogenic response
- constriction or dilation in the walls of arterioles in response to pressures related to blood flow; reduces high blood flow or increases low blood flow to help maintain consistent flow to the capillary network
- neurogenic shock
- type of shock that occurs with cranial or high spinal injuries that damage the cardiovascular centers in the medulla oblongata or the nervous fibers originating from this region
- obstructive shock
- type of shock that occurs when a significant portion of the vascular system is blocked
- sepsis
- (also, septicemia) organismal-level inflammatory response to a massive infection
- septic shock
- (also, blood poisoning) type of shock that follows a massive infection resulting in organism-wide inflammation
- vascular shock
- type of shock that occurs when arterioles lose their normal muscular tone and dilate dramatically


6.17: Earth and Impacts (Exercises)
For Further Exploration
Articles
Earth
Collins, W., et al. “The Physical Science behind Climate Change.” Scientific American (August 2007): 64. Why scientists are now confident that human activities are changing our planet’s climate.
Glatzmaier, G., & Olson, P. “Probing the Geodynamo.” Scientific American (April 2005): 50. Experiments and modeling that tell us about the source and reversals of Earth’s magnetic field.
Gurnis, M. “Sculpting the Earth from Inside Out.” Scientific American (March 2001): 40. On motions that lift and lower the continents.
Hartmann, W. “Piecing Together Earth’s Early History.” Astronomy (June 1989): 24.
Jewitt, D., & Young, E. “Oceans from the Skies.” Scientific American (March 2015): 36. How did Earth get its water after its initial hot period?
Impacts
Boslaugh, M. “In Search of Death-Plunge Asteroids.” Astronomy ( July 2015): 28. On existing and proposed programs to search for earth-crossing asteroids.
Brusatte, S. “What Killed the Dinosaurs?” Scientific American (December 2015): 54. The asteroid hit Earth at an already vulnerable time.
Chyba, C. “Death from the Sky: Tunguska.” Astronomy (December 1993): 38. Excellent review article.
Durda, D. “The Chelyabinsk Super-Meteor.” Sky & Telescope (June 2013): 24. A nice summary with photos and eyewitness reporting.
Gasperini, L., et al. “The Tunguska Mystery.” Scientific American (June 2008): 80. A more detailed exploration of the site of the 1908 impact over Siberia.
Kring, D. “Blast from the Past.” Astronomy (August 2006): 46. Six-page introduction to Arizona’s meteor crater.
Websites
Earth
Astronaut Photography of Earth from Space: earth.jsc.nasa.gov/. A site with many images and good information.
Exploration of the Earth’s Magnetosphere: http://phy6.org/Education/Intro.html . An educational website by Dr. Daniel Stern.
NASA Goddard: Earth from Space: Fifteen Amazing Things in 15 Years: https://www.nasa.gov/content/goddard...gs-in-15-years . Images and videos that reveal things about our planet and its atmosphere.
U.S. Geological Survey: Earthquake Information Center: http://earthquake.usgs.gov/learn/
Views of the Solar System: http://www.solarviews.com/eng/earth.htm . Overview of Earth.
Impacts
B612 Foundation : https://b612foundation.org/ . Set up by several astronauts for research and education about the asteroid threat to Earth and to build a telescope in space to search for dangerous asteroids.
Lunar and Planetary Institute: Introduction to Terrestrial Impact Craters: http://www.lpi.usra.edu/publications...esets/craters/ . Includes images.
Meteor Crater Tourist Site: http://meteorcrater.com/ .
NASA/Jet Propulsion Lab Near Earth Object Program: http://neo.jpl.nasa.gov/neo/ .
What Are Near-Earth-Objects: http://spaceguardcentre.com/what-are-neos/ . From the British Spaceguard Centre.
Videos
Earth
All Alone in the Night: http://apod.nasa.gov/apod/ap120305.html . Flying over Earth at night (2:30).
Earth Globes Movies (including Earth at night): astro.uchicago.edu/cosmus/projects/earth/.
PBS NOVA Videos about Earth: http://www.pbs.org/wgbh/nova/earth/ . Programs and information about planet Earth. Click full episodes on the menu at left to be taken to a nice array of videos.
U. S. National Weather Service: http://earth.nullschool.net . Real Time Globe of Earth showing wind patterns which can be zoomed and moved to your preferred view.
Impacts
Chelyabinsk Meteor: Can We Survive a Bigger Impact?: https://www.youtube.com/watch?v=Y-e6xyUZLLs . Talk by Dr. David Morrison (1:34:48).
Large Asteroid Impact Simulation: https://www.youtube.com/watch?v=bU1QPtOZQZU . Large asteroid impact simulation from the Discovery Channel (4:45).
Meteor Hits Russia February 15, 2013: https://www.youtube.com/watch?v=dpmXyJrs7iU . Archive of eyewitness footage (10:11).
Sentinel Mission: Finding an Asteroid Headed for Earth: https://www.youtube.com/watch?v=efz8c3ijD_A . Public lecture by astronaut Ed Lu (1:08:57).
Review Questions
- What is the thickest interior layer of Earth? The thinnest?
- What are Earth’s core and mantle made of? Explain how we know.
-
Explain briefly how the following phenomena happen on Earth, relating your answers to the theory of plate tectonics:
- earthquakes
- continental drift
- mountain building
- volcanic eruptions
- creation of the Hawaiian island chain
- What is the source of Earth’s magnetic field?
- Why is the shape of the magnetosphere not spherical like the shape of Earth?
- Although he did not present a mechanism, what were the key points of Alfred Wegener’s proposal for the concept of continental drift?
- List the possible interactions between Earth’s crustal plates that can occur at their boundaries.
- List, in order of decreasing altitude, the principle layers of Earth’s atmosphere.
- In which atmospheric layer are almost all water-based clouds formed?
- What is, by far, the most abundant component of Earth’s atmosphere?
- Describe three ways in which the presence of life has affected the composition of Earth’s atmosphere.
- Briefly describe the greenhouse effect.
- How do impacts by comets and asteroids influence Earth’s geology, its atmosphere, and the evolution of life?
- Why are there so many impact craters on our neighbor world, the Moon, and so few on Earth?
Thought Questions
- If you wanted to live where the chances of a destructive earthquake were small, would you pick a location near a fault zone, near a mid ocean ridge, near a subduction zone, or on a volcanic island such as Hawaii? What are the relative risks of earthquakes at each of these locations?
- Which type of object would likely cause more damage if it struck near an urban area: a small metallic object or a large stony/icy one?
- If all life were destroyed on Earth by a large impact, would new life eventually form to take its place? Explain how conditions would have to change for life to start again on our planet.
- Why is a decrease in Earth’s ozone harmful to life?
- Why are we concerned about the increases in CO 2 and other gases that cause the greenhouse effect in Earth’s atmosphere? What steps can we take in the future to reduce the levels of CO2 in our atmosphere? What factors stand in the way of taking the steps you suggest? (You may include technological, economic, and political factors in your answer.)
- Do you think scientists should make plans to defend Earth from future asteroid impacts? Is it right to intervene in the same evolutionary process that made the development of mammals (including us) possible after the big impact 65 million years ago?


5.8: The Chicago School
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- Alison S. Burke, David Carter, Brian Fedorek, Tiffany Morey, Lore Rutz-Burri, & Shanell Sanchez
- Southern Oregon University via OpenOregon
Biological and psychological positivism looked at differences between criminals and non-criminals. Instead of finding differences between kinds of people, the Chicago School tried to detect differences between kinds of places. During the 1920-1930s, the University of Chicago was the vanguard for human ecology, which is the study of the relationship between humans and their environment. Robert Park (1925) viewed cities as “super-organisms,” comparing the city-human relationship to the natural ecosystems of plants and animals that share habitats. Furthermore, Burgess (1925) proposed concentric zone theory, which explained how cities grow – from the central business district outwards. [1]
Shaw and McKay (1942), who were both former students of Burgess, began to plot the addresses of juvenile court-referred male youths. They noticed many of the addresses were located in the zone in transition. [2] Upon further investigation, Shaw and McKay noticed three qualitative differences in the transitional zone compared to other zones. First, the physical status included the invasion of industry and the largest number of condemned buildings. When many buildings are in disrepair, population levels decrease. Second, the population composition was also different. The zone in transition had higher concentrations of foreign-born and African-American heads of families. It also had a transient population. Third, the transitional zone had socioeconomic differences with the highest rates of welfare, lowest median rent, the lowest percentage of family-owned houses. Interrelated, the zone also had the highest rates of infant deaths, tuberculosis, and mental illness.
Shaw and McKay believed the zone in transition led to social disorganization. Social disorganization is the inability of social institutions to control an individual’s behavior. Since the zone in transition had people moving in and moving out at such high rates, social institutions (like family, school, religion, government, economy) and members could no longer agree on essential norms and values. As earlier stated, many residents were foreign. Thus, speaking different languages and having different religious beliefs may have prevented neighbors from talking to one another and solidifying community bonds. Overall, Shaw and McKay were two of the first theorists to put forth the premise that community characteristics matter when discussing criminal behavior.


19.2: The African American “Great Migration” and New European Immigration
New cities were populated with diverse waves of new arrivals, who came to the cities to seek work in the businesses and factories there. While a small percentage of these newcomers were white Americans seeking jobs, most were made up of two groups that had not previously been factors in the urbanization movement: African Americans fleeing the racism of the farms and former plantations in the South, and southern and eastern European immigrants. These new immigrants supplanted the previous waves of northern and western European immigrants, who had tended to move west to purchase land. Unlike their predecessors, the newer immigrants lacked the funds to strike out to the western lands and instead remained in the urban centers where they arrived, seeking any work that would keep them alive.
THE AFRICAN AMERICAN “GREAT MIGRATION”
Between the end of the Civil War and the beginning of the Great Depression, nearly two million African Americans fled the rural South to seek new opportunities elsewhere. While some moved west, the vast majority of this Great Migration, as the large exodus of African Americans leaving the South in the early twentieth century was called, traveled to the Northeast and Upper Midwest. The following cities were the primary destinations for these African Americans: New York, Chicago, Philadelphia, St. Louis, Detroit, Pittsburgh, Cleveland, and Indianapolis. These eight cities accounted for over two-thirds of the total population of the African American migration.
A combination of both “push” and “pull” factors played a role in this movement. Despite the end of the Civil War and the passage of the Thirteenth, Fourteenth, and Fifteenth Amendments to the U.S. Constitution (ensuring freedom, the right to vote regardless of race, and equal protection under the law, respectively), African Americans were still subjected to intense racial hatred. The rise of the Ku Klux Klan in the immediate aftermath of the Civil War led to increased death threats, violence, and a wave of lynchings. Even after the formal dismantling of the Klan in the late 1870s, racially motivated violence continued. According to researchers at the Tuskegee Institute, there were thirty-five hundred racially motivated lynchings and other murders committed in the South between 1865 and 1900. For African Americans fleeing this culture of violence, northern and midwestern cities offered an opportunity to escape the dangers of the South.
In addition to this “push” out of the South, African Americans were also “pulled” to the cities by factors that attracted them, including job opportunities, where they could earn a wage rather than be tied to a landlord, and the chance to vote (for men, at least), supposedly free from the threat of violence. Although many lacked the funds to move themselves north, factory owners and other businesses that sought cheap labor assisted the migration. Often, the men moved first then sent for their families once they were ensconced in their new city life. Racism and a lack of formal education relegated these African American workers to many of the lower-paying unskilled or semi-skilled occupations. More than 80 percent of African American men worked menial jobs in steel mills, mines, construction, and meat packing. In the railroad industry, they were often employed as porters or servants (Figure 19.2.1). In other businesses, they worked as janitors, waiters, or cooks. African American women, who faced discrimination due to both their race and gender, found a few job opportunities in the garment industry or laundries, but were more often employed as maids and domestic servants. Regardless of the status of their jobs, however, African Americans earned higher wages in the North than they did for the same occupations in the South, and typically found housing to be more available.
However, such economic gains were offset by the higher cost of living in the North, especially in terms of rent, food costs, and other essentials. As a result, African Americans often found themselves living in overcrowded, unsanitary conditions, much like the tenement slums in which European immigrants lived in the cities. For newly arrived African Americans, even those who sought out the cities for the opportunities they provided, life in these urban centers was exceedingly difficult. They quickly learned that racial discrimination did not end at the Mason-Dixon Line, but continued to flourish in the North as well as the South. European immigrants, also seeking a better life in the cities of the United States, resented the arrival of the African Americans, whom they feared would compete for the same jobs or offer to work at lower wages. Landlords frequently discriminated against them; their rapid influx into the cities created severe housing shortages and even more overcrowded tenements. Homeowners in traditionally white neighborhoods later entered into covenants in which they agreed not to sell to African American buyers; they also often fled neighborhoods into which African Americans had gained successful entry. In addition, some bankers practiced mortgage discrimination, later known as “redlining,” in order to deny home loans to qualified buyers. Such pervasive discrimination led to a concentration of African Americans in some of the worst slum areas of most major metropolitan cities, a problem that remained ongoing throughout most of the twentieth century.
So why move to the North, given that the economic challenges they faced were similar to those that African Americans encountered in the South? The answer lies in noneconomic gains. Greater educational opportunities and more expansive personal freedoms mattered greatly to the African Americans who made the trek northward during the Great Migration. State legislatures and local school districts allocated more funds for the education of both blacks and whites in the North, and also enforced compulsory school attendance laws more rigorously. Similarly, unlike the South where a simple gesture (or lack of a deferential one) could result in physical harm to the African American who committed it, life in larger, crowded northern urban centers permitted a degree of anonymity—and with it, personal freedom—that enabled African Americans to move, work, and speak without deferring to every white person with whom they crossed paths. Psychologically, these gains more than offset the continued economic challenges that black migrants faced.
THE CHANGING NATURE OF EUROPEAN IMMIGRATION
Immigrants also shifted the demographics of the rapidly growing cities. Although immigration had always been a force of change in the United States, it took on a new character in the late nineteenth century. Beginning in the 1880s, the arrival of immigrants from mostly southern and eastern European countries rapidly increased while the flow from northern and western Europe remained relatively constant (Table 19.2.1).
| Table 19.2.1 : Cumulative Total of the Foreign-Born Population in the United States, 1870–1910 (by major country of birth and European region) | |||||
|---|---|---|---|---|---|
| Region Country | 1870 | 1880 | 1890 | 1900 | 1910 |
| Northern and Western Europe | 4,845,679 | 5,499,889 | 7,288,917 | 7,204,649 | 7,306,325 |
| Germany | 1,690,533 | 1,966,742 | 2,784,894 | 2,663,418 | 2,311,237 |
| Ireland | 1,855,827 | 1,854,571 | 1,871,509 | 1,615,459 | 1,352,251 |
| England | 550,924 | 662,676 | 908,141 | 840,513 | 877,719 |
| Sweden | 97,332 | 194,337 | 478,041 | 582,014 | 665,207 |
| Austria | 30,508 | 38,663 | 123,271 | 275,907 | 626,341 |
| Norway | 114,246 | 181,729 | 322,665 | 336,388 | 403,877 |
| Scotland | 140,835 | 170,136 | 242,231 | 233,524 | 261,076 |
| Southern and Eastern Europe | 93,824 | 248,620 | 728,851 | 1,674,648 | 4,500,932 |
| Italy | 17,157 | 44,230 | 182,580 | 484,027 | 1,343,125 |
| Russia | 4,644 | 35,722 | 182,644 | 423,726 | 1,184,412 |
| Poland | 14,436 | 48,557 | 147,440 | 383,407 | 937,884 |
| Hungary | 3,737 | 11,526 | 62,435 | 145,714 | 495,609 |
| Czechoslovakia | 40,289 | 85,361 | 118,106 | 156,891 | 219,214 |
The previous waves of immigrants from northern and western Europe, particularly Germany, Great Britain, and the Nordic countries, were relatively well off, arriving in the country with some funds and often moving to the newly settled western territories. In contrast, the newer immigrants from southern and eastern European countries, including Italy, Greece, and several Slavic countries including Russia, came over due to “push” and “pull” factors similar to those that influenced the African Americans arriving from the South. Many were “pushed” from their countries by a series of ongoing famines, by the need to escape religious, political, or racial persecution, or by the desire to avoid compulsory military service. They were also “pulled” by the promise of consistent, wage-earning work.
Whatever the reason, these immigrants arrived without the education and finances of the earlier waves of immigrants, and settled more readily in the port towns where they arrived, rather than setting out to seek their fortunes in the West. By 1890, over 80 percent of the population of New York would be either foreign-born or children of foreign-born parentage. Other cities saw huge spikes in foreign populations as well, though not to the same degree, due in large part to Ellis Island in New York City being the primary port of entry for most European immigrants arriving in the United States.
The number of immigrants peaked between 1900 and 1910, when over nine million people arrived in the United States. To assist in the processing and management of this massive wave of immigrants, the Bureau of Immigration in New York City, which had become the official port of entry, opened Ellis Island in 1892. Today, nearly half of all Americans have ancestors who, at some point in time, entered the country through the portal at Ellis Island. Doctors or nurses inspected the immigrants upon arrival, looking for any signs of infectious diseases (Figure 19.2.2). Most immigrants were admitted to the country with only a cursory glance at any other paperwork. Roughly 2 percent of the arriving immigrants were denied entry due to a medical condition or criminal history. The rest would enter the country by way of the streets of New York, many unable to speak English and totally reliant on finding those who spoke their native tongue.
Seeking comfort in a strange land, as well as a common language, many immigrants sought out relatives, friends, former neighbors, townspeople, and countrymen who had already settled in American cities. This led to a rise in ethnic enclaves within the larger city. Little Italy, Chinatown, and many other communities developed in which immigrant groups could find everything to remind them of home, from local language newspapers to ethnic food stores. While these enclaves provided a sense of community to their members, they added to the problems of urban congestion, particularly in the poorest slums where immigrants could afford housing.
This Library of Congress exhibit on the history of Jewish immigration to the United States illustrates the ongoing challenge immigrants felt between the ties to their old land and a love for America.
The demographic shift at the turn of the century was later confirmed by the Dillingham Commission, created by Congress in 1907 to report on the nature of immigration in America; the commission reinforced this ethnic identification of immigrants and their simultaneous discrimination. The report put it simply: These newer immigrants looked and acted differently. They had darker skin tone, spoke languages with which most Americans were unfamiliar, and practiced unfamiliar religions, specifically Judaism and Catholicism. Even the foods they sought out at butchers and grocery stores set immigrants apart. Because of these easily identifiable differences, new immigrants became easy targets for hatred and discrimination. If jobs were hard to find, or if housing was overcrowded, it became easy to blame the immigrants. Like African Americans, immigrants in cities were blamed for the problems of the day.
Growing numbers of Americans resented the waves of new immigrants, resulting in a backlash. The Reverend Josiah Strong fueled the hatred and discrimination in his bestselling book, Our Country: Its Possible Future and Its Present Crisis , published in 1885. In a revised edition that reflected the 1890 census records, he clearly identified undesirable immigrants—those from southern and eastern European countries—as a key threat to the moral fiber of the country, and urged all good Americans to face the challenge. Several thousand Americans answered his call by forming the American Protective Association, the chief political activist group to promote legislation curbing immigration into the United States. The group successfully lobbied Congress to adopt both an English language literacy test for immigrants, which eventually passed in 1917, and the Chinese Exclusion Act (discussed in a previous chapter). The group’s political lobbying also laid the groundwork for the subsequent Emergency Quota Act of 1921 and the Immigration Act of 1924, as well as the National Origins Act.
The global timeline of immigration at the Library of Congress offers a summary of immigration policies and the groups affected by it, as well as a compelling overview of different ethnic groups’ immigration stories. Browse through to see how different ethnic groups made their way in the United States.
Section Summary
For both African Americans migrating from the postwar South and immigrants arriving from southeastern Europe, a combination of “push” and “pull” factors influenced their migration to America’s urban centers. African Americans moved away from the racial violence and limited opportunities that existed in the rural South, seeking wages and steady work, as well as the opportunity to vote safely as free men; however, they quickly learned that racial discrimination and violence were not limited to the South. For European immigrants, famine and persecution led them to seek a new life in the United States, where, the stories said, the streets were paved in gold. Of course, in northeastern and midwestern cities, both groups found a more challenging welcome than they had anticipated. City residents blamed recent arrivals for the ills of the cities, from overcrowding to a rise in crime. Activist groups pushed for anti-immigration legislation, seeking to limit the waves of immigrants that sought a better future in the United States.
Review Questions
Why did African Americans consider moving from the rural South to the urban North following the Civil War?
to be able to buy land
to avoid slavery
to find wage-earning work
to further their education
C
Which of the following is true of late nineteenth-century southern and eastern European immigrants, as opposed to their western and northern European predecessors?
Southern and eastern European immigrants tended to be wealthier.
Southern and eastern European immigrants were, on the whole, more skilled and able to find better paying employment.
Many southern and eastern European immigrants acquired land in the West, while western and northern European immigrants tended to remain in urban centers.
Ellis Island was the first destination for most southern and eastern Europeans.
D
What made recent European immigrants the ready targets of more established city dwellers? What was the result of this discrimination?
Newer immigrants often had different appearances, spoke unfamiliar languages, and lived their lives—from the religions they practiced to the food they ate—in ways that were alien to many Americans. In all of city life’s more challenging aspects, from competition for jobs to overcrowding in scarce housing, immigrants became easy scapegoats. The Reverend Josiah Strong’s bestselling book, Our Country: Its Possible Future and Its Present Crisis , fueled this discrimination. The American Protective Association, the chief political activist group promoting anti-immigration legislation, formed largely in response to Strong’s call.
Glossary
- Great Migration
- the name for the large wave of African Americans who left the South after the Civil War, mostly moving to cities in the Northeast and Upper Midwest


6.5: Bayes' Formula
Learning Objectives
In this section, you will learn to:
- Find probabilities using Bayes’ formula.
- Use a probability tree to find and represent values needed when using Bayes’ formula.
Prerequisite Skills
Before you get started, take this prerequisite quiz.
1. If \(P(F) = .4\), \(P(E | F) = .3\), find \(P\)(\(E \cap F\)).
- Click here to check your answer
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\(.12\)
If you missed this problem, review Section 6.4 . (Note that this will open in a new window.)
2. \(P(E) = .3\), \(P(F) = .3\); \(E \cap F\) are mutually exclusive. Find \(P(E | F)\).
- Click here to check your answer
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\(.3\)
If you missed this problem, review Section 6.4 . (Note that this will open in a new window.)
3. If \(P(E) = .6\), \(P\)(\(E \cap F\)) = .24, find \(P(F | E)\).
- Click here to check your answer
-
\(.4\)
If you missed this problem, review Section 6.4 . (Note that this will open in a new window.)
4. If \(P(E \cap F\)) = \(.04\) , \(P(E | F) = .1\), find \(P(F)\).
- Click here to check your answer
-
\(.4\)
If you missed this problem, review Section 6.4 . (Note that this will open in a new window.)
In this section, we will develop and use Bayes' Formula to solve an important type of probability problem. Bayes' formula is a method of calculating the conditional probability \(P(F | E)\) from \(P(E | F)\). The ideas involved here are not new, and most of these problems can be solved using a tree diagram. However, Bayes' formula does provide us with a tool with which we can solve these problems without a tree diagram.
We begin with an example.
Example \(\PageIndex{1}\)
Suppose you are given two jars.  A jar and a marble are selected at random. Construct a tree diagram to represent this scenario and to answer the following questions.
- What is the probability that the marble chosen is a black marble?
- If the chosen marble is black, what is the probability that it came from Jar I?
- If the chosen marble is black, what is the probability that it came from Jar II?
Solution
Let \(J I\) be the event that Jar I is chosen, \(J II\) be the event that Jar II is chosen, \(B\) be the event that a black marble is chosen and \(W\) the event that a white marble is chosen.
We illustrate using a tree diagram.
- The probability that a black marble is chosen is \(P(B)\) = 1/10 + 2/10 = 3/10.
- To find \(P(J I | B)\), we use the definition of conditional probability, and we get \[P(J I | B)=\frac{P(J I \cap B)}{P(B)}=\frac{1 / 10}{3 / 10}=\frac{1}{3} \nonumber\]
- Similarly, \(\mathrm{P}(\mathrm{J} \mathrm{II} | \mathrm{B})=\frac{\mathrm{P}(\mathrm{J} \mathrm{II} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})}=\frac{2 / 10}{3 / 10}=\frac{2}{3}\)
In parts b and c, the reader should note that the denominator is the sum of all probabilities of all branches of the tree that produce a black marble, while the numerator is the branch that is associated with the particular jar in question.
We will soon discover that this is a statement of Bayes' formula.
Let us first visualize the problem.
We are given a sample space \(\mathrm{S}\) and two mutually exclusive events \(J I\) and \(J II\). That is, the two events, \(J I\) and \(J II\), divide the sample space into two parts such that \(\mathrm{JI} \cup \mathrm{JII}=\mathrm{S}\). Furthermore, we are given an event \(B\) that has elements in both \(J I\) and \(J II\), as shown in the Venn diagram below.
From the Venn diagram, we can see that \(\mathrm{B}=(\mathrm{B} \cap \mathrm{J} \mathrm{I}) \cup(\mathrm{B} \cap \mathrm{J} \mathrm{II})\) Therefore:
\[ P(B)=P(B \cap J I)+P(B \cap J I I) \nonumber \]
But the multiplication rule in Section 6.5 gives us
\[ P(B \cap J I)=P(J I) \cdot P(B | J I) \quad \text { and } \quad P(B \cap J I I)=P(J I I) \cdot P(B | J I I) \nonumber\]
By substitution, we get
\[P(B)=P(J I) \cdot P(B | J I)+P(J I I) \cdot P(B | J I I) \nonumber\]
The conditional probability formula gives us
\[P(J I | B)=\frac{P(J I \cap B)}{P(B)} \nonumber\]
Therefore, \(P(J I | B)=\frac{P(J I) \cdot P(B | J I)}{P(B)}\)
or
\[P(J I | B)=\frac{P(J I) \cdot P(B | J I)}{P(J I) \cdot P(B | J I)+P(J I I) \cdot P(B | J I I)} \nonumber\]
The last statement is Bayes' Formula for the case where the sample space is divided into two partitions.
The following is the generalization of Bayes’ formula for n partitions.
Bayes' Formula for \(n\) partitions
Let \(\mathrm{S}\) be a sample space that is divided into \(n\) partitions, \(A_1\), \(A_2\), . . . \(A_n\). If \(E\) is any event in \(\mathrm{S}\), then
\[\mathbf{P}\left(\mathbf{A}_{\mathbf{i}} | \mathbf{E}\right)=\frac{\mathbf{P}\left(\mathbf{E} | \mathbf{A}_{\mathbf{i}}\right)\mathbf{P}\left(\mathbf{A}_{\mathbf{i}}\right)}{\mathbf{P}\left(\mathbf{E} | \mathbf{A}_{\mathbf{1}}\right)\mathbf{P}\left(\mathbf{A}_{\mathbf{1}}\right) +\mathbf{P}\left(\mathbf{E} | \mathbf{A}_{2}\right)\mathbf{P}\left(\mathbf{A}_{2}\right) +\cdots+\mathbf{P}\left(\mathbf{E} | \mathbf{A}_{\mathbf{n}}\right)\mathbf{P}\left(\mathbf{A}_{\mathbf{n}}\right) } \nonumber\]
Note that this can be summarized as \[\mathbf{P}\left(\mathbf{A}_{\mathbf{i}} | \mathbf{E}\right)=\frac{\text{product of branches leading to E through } \mathbf{A}_{\mathbf{i}}}{\text{sum of branch products leading to E}} \nonumber \]
We begin with the following example.
Example \(\PageIndex{2}\)
A department store buys 50% of its appliances from Manufacturer A, 30% from Manufacturer B, and 20% from Manufacturer C. It is estimated that 6% of Manufacturer A's appliances, 5% of Manufacturer B's appliances, and 4% of Manufacturer C's appliances need repair before the warranty expires. An appliance is chosen at random. If the appliance chosen needed repair before the warranty expired, what is the probability that the appliance was manufactured by Manufacturer A? Manufacturer B? Manufacturer C?
Solution
Let A, B and C be the events that the appliance is manufactured by Manufacturer A, Manufacturer B, and Manufacturer C, respectively. Further, suppose that the event R denotes that the appliance needs repair before the warranty expires.
We need to find P(A | R), P(B | R) and P(C | R).
We will do this problem both by using a tree diagram and by using Bayes' formula.
We draw a tree diagram.
The probability P(A | R), for example, is a fraction whose denominator is the sum of all probabilities of all branches of the tree that result in an appliance that needs repair before the warranty expires, and the numerator is the branch that is associated with Manufacturer A. P(B | R) and P(C | R) are found in the same way.
\[\begin{array}{l}
P(A | R)=\frac{.030}{(.030)+(.015)+(.008)}=\frac{.030}{.053}=.566 \\
P(B | R)=\frac{.015}{.053}=.283 \text { and } P(C | R)=\frac{.008}{.053}=.151
\end{array} \nonumber\]
Alternatively, using Bayes' formula,
\begin{aligned}
\mathrm{P}(\mathrm{A} | \mathrm{R}) &=\frac{\mathrm{P}(\mathrm{R} | \mathrm{A})\mathrm{P}(\mathrm{A}) }{\mathrm{P}(\mathrm{R} | \mathrm{A})\mathrm{P}(\mathrm{A}) + \mathrm{P}(\mathrm{R} | \mathrm{B})\mathrm{P}(\mathrm{B}) + \mathrm{P}(\mathrm{R} | \mathrm{C})\mathrm{P}(\mathrm{C}) } \\
&=\frac{.030}{(.030)+(.015)+(.008)}=\frac{.030}{.053}=.566
\end{aligned}
P(B | R) and P(C | R) can be determined in the same manner.
P(B | R) = \(\frac{.015}{.053}=.283\) and P(C | R) = \(\frac{.008}{.053}=.151\).
Example \(\PageIndex{3}\)
There are five American Furniture Warehouse stores in the Denver area. The percent of employees over the age of 50 is given in the table below.
|
Store Number |
Number of Employees |
Percent of 50+ Employees |
|
1 |
300 |
.40 |
|
2 |
150 |
.65 |
|
3 |
200 |
.60 |
|
4 |
250 |
.50 |
|
5 |
100 |
.70 |
|
Total = 1000 |
If an employee chosen at random is over 50 years old, what is the probability that the employee works at store III?
Solution
Let \(k\) = 1, 2, . . . , 5 be the event that the employee worked at store \(k\), and A be the event that the employee is over 50 years old. Since there are a total of 1000 employees at the five stores,
\[P(1)=.30 \quad P(2)=.15 \quad P(3)=.20 \quad P(4)=.25 \quad P(5)=.10 \nonumber\]
Using Bayes' formula,
\[\begin{array}{l}
\mathrm{P}(3 | \mathrm{A})&=\frac{\mathrm{P}(\mathrm{A} | 3)\mathrm{P}(3) }{\mathrm{P}(\mathrm{A} | 1)\mathrm{P}(1) + \mathrm{P}(\mathrm{A} | 2)\mathrm{P}(2) + \mathrm{P}(\mathrm{A} | 3)\mathrm{P}(3) + \mathrm{P}(\mathrm{A} | 4)\mathrm{P}(4) + \mathrm{P}(\mathrm{A} | 5)\mathrm{P}(5) } \\
&=\frac{(.60)(.20)}{(.40)(.30)+(.65)(.15)+(.60)(.20)+(.50)(.25)+(.70)(.10)} \\
&=.2254
\end{array} \nonumber\]


3.7: The Projection Operator
Learning Objectives
- Explain the projection operator
A frequent source of confusion in relativity is that we write down equations that are coordinate-dependent, but forget the dependency. Similarly, it is possible to write expressions that are only valid for one choice of signature. The following notation, defining a projection operator \(P\), is one tool for avoiding these difficulties.
\[P_or = r - \dfrac{r\cdot o}{o\cdot o}o\]
Usually \(o\) is the future timelike vector representing a certain observer, but the definition can be applied as long as \(o\) isn’t lightlike. The idea being expressed is that we want to get rid of any part of \(r\) that is parallel to \(o\)’s arrow of time. In a graph constructed according to \(o\)’s Minkowski coordinates, we cast \(r\)’s shadow down perpendicularly onto the spacelike axis, or the spacelike three-plane in \(3+1\) dimensions. This is why \(P\) is referred to as a projection operator. The notation sometimes allows us to express the things that we would otherwise express by explicitly or implicitly constructing and referring to \(o\)’s spacelike Minkowski coordinates.
Properties of projection operator
\(P\) has the following properties:
- \(o\cdot P_or = 0\)
- \(r - P_or\) is parallel to \(o\).
- \(P_o o = 0\)
- \(P_o P_or = P_or\)
- \(P_{co} = P_o\)
- \(P_o\) is linear, i.e., \(P_o(q +r) = P_oq + P_or\) and \(P_o(cr) = cP_or\)
- \(\tfrac{d}{dx}P_or = P_o\tfrac{dr}{dx}\), where \(x\) is any variable and \(o\) doesn’t depend on \(x\).
- If \(o\) and \(v\) are both future timelike, and \(|o^2| = 1\), then we can express \(v\) as \(v = P_ov + \gamma o\), where \(\gamma\) has the usual interpretation for world-lines that coincide with these two vectors.
All of these hold regardless of whether the signature is \(+---\) or \(-+ ++\), and none of them refer to any coordinates. Properties 1 and 2 can serve as an alternative, geometrical definition of \(P\). Property 3 says that an observer considers herself to be at rest. Property 4 is a general property of all projection operators. Property 8 splits the vector into its spatial and temporal parts according to \(o\).
Sometimes if we know a position, velocity, or acceleration fourvector, we want to find out how these would be measured by a particular observer using clocks and rulers. The following table \(\PageIndex{1}\) shows how to switch back and forth between the two representations. We use, for example, the notation \(v_o\) to mean the velocity vector of the form \((0,v_x,v_y,v_z)\) that would be measured by an observer whose velocity vector is \(o\) (so that the subscript is an “\(o\)” for “observer,” not a zero). Since this type of vector, expressed in the Minkowski coordinates of observer \(o\), has a zero time component, we refer to it as a three-vector. In all of these expressions, the velocity vectors \(o\) and \(v\) are assumed to be normalized, and the signature is assumed to be \(+---\) (one implication being that \(o\cdot v\) is simply \(\gamma\)).
| Finding the three-vector from the four-vector | Finding the four-vector from the three-vector |
|---|---|
| \(X_o = P_oX\) | |
| \(v_o = \dfrac{P_ov}{o\cdot v}\) | \(v = \gamma (o + v_o)\) |
| \(a_o = \dfrac{1}{(o\cdot v)^2}\left [ P_oa - (o\cdot a)v_o \right ]\) | \(a = \gamma ^3(a_o\cdot v_o)v + \gamma ^2a_o\), where \(v\) is found as above |
As an example of how these are derived, the three-velocity \(v_o\) is the derivative of \(x_o\) with respect to observer \(o\)’s Minkowski time coordinate \(t\), whereas the four-velocity is defined as the derivative of \(x\) with respect to the proper time \(τ\) of the world-line being observed. Therefore we have
\[\begin{align*} v_o &= \dfrac{\mathrm{dX_o} }{\mathrm{d} t}\\[5pt] &= \dfrac{\mathrm{d} P_oX}{\mathrm{d} t} \end{align*}\]
and applying property 7 of the projection operator this becomes
\[\begin{align*} v_o &= P_o\dfrac{\mathrm{dX} }{\mathrm{d} t}\\[5pt] &= P_o\dfrac{\mathrm{d} X}{\mathrm{d} \tau }\dfrac{\mathrm{d} \tau }{\mathrm{d} t}\\[5pt] &= \dfrac{1}{\gamma }P_o\dfrac{\mathrm{d} X}{\mathrm{d} \tau }\\[5pt] &= \dfrac{1}{o\cdot v}P_o\dfrac{\mathrm{d} X}{\mathrm{d} \tau }\\[5pt] &= \dfrac{P_o v}{o\cdot v} \end{align*}\]
The similar but messier derivation of the expression for \(a_o\) is problem Q15. In manipulating expressions of this type, the identity \(\dfrac{\mathrm{d} \gamma }{\mathrm{d} t} = \gamma ^3 a_o\cdot v_o\) is often handy.
Example \(\PageIndex{1}\): Lewis-Tolman paradox
The following example is a form of a paradox discussed by Lewis and Tolman in 1909.
Figure \(\PageIndex{1}\) shows the frame of reference of observer \(o\) in which identical particles \(1\) and \(2\) are at initially rest and located at equal distances \(l\) from the origin along the \(y\) and \(x\) axes. External forces of equal strength act in the directions shown by the arrows so as to produce accelerations of magnitude \(α\). The system is in rotational equilibrium \(dL/dt = 0\), because the rate at which particle \(1\) picks up clockwise angular momentum is the same as the rate at which \(2\) acquires it in the counterclockwise direction.
Now change to the frame of reference o0, moving to the right relative to \(o\) at velocity \(v\). Particle \(2\)’s distance from the origin is Lorentz-contracted from \(l\) to \(l/\gamma\), so its angular momentum is also reduced by \(1/\gamma\). It now appears that the system’s total angular momentum is increasing in the clockwise sense. How can we have rotational equilibrium in one frame, but not another?
The resolution of the paradox is that the accelerations transform as well. In the original frame \(o\), the four-velocities are \(v_1 = v_2 = (1,0,0,0)\), and the four-accelerations are \(a_1 = (0,α,0,0)\) and \(a_2 = (0,0,α,0)\). Applying a Lorentz transformation, we have \(v_{1}^{'} = v_{2}^{'} = (\gamma ,-\gamma v,0,0)\) and
\[a_{1}^{'} = \alpha (-\gamma v,\gamma ,0,0)\\[5pt] a_{2}^{'} = \alpha (0,0,1,0) \nonumber\]
Our definition of angular momentum is expressed in terms of three-vectors such as \(a_{o'1}\) and \(a_{o'2}\), not four-vectors like \(a_{1}^{'}\) and \(a_{2}^{'}\). We have
\[\dfrac{\mathrm{d} L'}{\mathrm{d} t'} = ma_{o'1x}l - ma_{o'2y}\dfrac{l}{\gamma } \nonumber\]
Using the relations \(v_o = \gamma ^{-1}P_o v\) and \(a_o = \gamma ^{-2}\left [ P_o a - (o\cdot a)v_o\right ]\), we find
\[v_{o'1x} = -v\]
\[a_{o'1x} = \dfrac{1}{\gamma ^2}\left [ \alpha \gamma - (-\alpha \gamma v)(-v) \right ] = \dfrac{\alpha }{\gamma ^3} \nonumber\]
and
\[a_{o'2y} = \dfrac{\alpha }{\gamma ^2} \nonumber\]
The result is
\[\dfrac{\mathrm{d} L'}{\mathrm{d} t'} = m\dfrac{\alpha }{\gamma ^3}l - m\dfrac{\alpha }{\gamma ^2}\dfrac{l}{\gamma } \nonumber\]
which is zero.


18.16: Waves
$Check Your Understanding
16.1. The wavelength of the waves depends on the frequency and the velocity of the wave. The frequency of the sound wave is equal to the frequency of the wave on the string. The wavelengths of the sound waves and the waves on the string are equal only if the velocities of the waves are the same, which is not always the case. If the speed of the sound wave is different from the speed of the wave on the string, the wavelengths are different. This velocity of sound waves will be discussed in Sound .
16.2. In a transverse wave, the wave may move at a constant propagation velocity through the medium, but the medium oscillates perpendicular to the motion of the wave. If the wave moves in the positive x-direction, the medium oscillates up and down in the y-direction. The velocity of the medium is therefore not constant, but the medium’s velocity and acceleration are similar to that of the simple harmonic motion of a mass on a spring.
16.3. Yes, a cosine function is equal to a sine function with a phase shift, and either function can be used in a wave function. Which function is more convenient to use depends on the initial conditions. In Figure 16.11 , the wave has an initial height of y(0.00, 0.00) = 0 and then the wave height increases to the maximum height at the crest. If the initial height at the initial time was equal to the amplitude of the wave y(0.00, 0.00) = +A, then it might be more convenient to model the wave with a cosine function.
16.4. This wave, with amplitude A = 0.5 m, wavelength \(\lambda\) = 10.00 m, period T = 0.50 s, is a solution to the wave equation with a wave velocity v = 20.00 m/s.
16.5. Since the speed of a wave on a taunt string is proportional to the square root of the tension divided by the linear density, the wave speed would increase by \(\sqrt{2}\).
16.6. At first glance, the time-averaged power of a sinusoidal wave on a string may look proportional to the linear density of the string because P = \(\frac{1}{2} \mu\)A 2 \(\omega^{2}\)v; however, the speed of the wave depends on the linear density. Replacing the wave speed with \(\sqrt{\frac{F_{T}}{\mu}}\) shows that the power is proportional to the square root of tension and proportional to the square root of the linear mass density: \[P = \frac{1}{2} \mu A^{2} \omega^{2} v = \frac{1}{2} \mu A^{2} \omega^{2} \sqrt{\frac{F_{T}}{\mu}} = \frac{1}{2} A^{2} \omega^{2} \sqrt{\mu F_{T}} \ldotp\]
16.7. Yes, the equations would work equally well for symmetric boundary conditions of a medium free to oscillate on each end where there was an antinode on each end. The normal modes of the first three modes are shown below. The dotted line shows the equilibrium position of the medium.
Note that the first mode is two quarters, or one half, of a wavelength. The second mode is one quarter of a wavelength, followed by one half of a wavelength, followed by one quarter of a wavelength, or one full wavelength. The third mode is one and a half wavelengths. These are the same result as the string with a node on each end. The equations for symmetrical boundary conditions work equally well for fixed boundary conditions and free boundary conditions. These results will be revisited in the next chapter when discussing sound wave in an open tube.
Conceptual Questions
1. A wave on a guitar string is an example of a transverse wave. The disturbance of the string moves perpendicular to the propagation of the wave. The sound produced by the string is a longitudinal wave where the disturbance of the air moves parallel to the propagation of the wave.
3. Propagation speed is the speed of the wave propagating through the medium. If the wave speed is constant, the speed can be found by v = \(\frac{\lambda}{T}\) = \(\lambda\)f. The frequency is the number of wave that pass a point per unit time. The wavelength is directly proportional to the wave speed and inversely proportional to the frequency.
5. No, the distance you move your hand up and down will determine the amplitude of the wave. The wavelength will depend on the frequency you move your hand up and down, and the speed of the wave through the spring.
7. Light from the Sun and stars reach Earth through empty space where there is no medium present.
9. The wavelength is equal to the velocity of the wave times the frequency and the wave number is equal to k = \(\frac{2 \pi}{\lambda}\), so yes, the wave number will depend on the frequency and also depend on the velocity of the wave propagating through the spring.
11. The medium moves in simple harmonic motion as the wave propagates through the medium, continuously changing speed, therefore it accelerates. The acceleration of the medium is due to the restoring force of the medium, which acts in the opposite direction of the displacement.
13. The wave speed is proportional to the square root of the tension, so the speed is doubled.
15. Since the speed of a wave on a string is inversely proportional to the square root of the linear mass density, the speed would be higher in the low linear mass density of the string.
17. The tension in the wire is due to the weight of the electrical power cable.
19. The time averaged power is P = \(\frac{E_{\lambda}}{T}\) = \(\frac{1}{2} \mu\)A 2 \(\omega^{2} \frac{\lambda}{T} = \frac{1}{2} \mu\)A 2 \(\omega^{2}\)v. If the frequency or amplitude is halved, the power decreases by a factor of 4.
21. As a portion on the string moves vertically, it exerts a force on the neighboring portion of the string, doing work on the portion and transferring the energy.
23. The intensity of a spherical wave is I = \(\frac{P}{4 \pi r^{2}}\), if no energy is dissipated the intensity will decrease by a factor of nine at three meters.
25. At the interface, the incident pulse produces a reflected pulse and a transmitted pulse. The reflected pulse would be out of phase with respect to the incident pulse, and would move at the same propagation speed as the incident pulse, but would move in the opposite direction. The transmitted pulse would travel in the same direction as the incident pulse, but at half the speed. The transmitted pulse would be in phase with the incident pulse. Both the reflected pulse and the transmitted pulse would have amplitudes less than the amplitude of the incident pulse.
27.
29. It may be as easy as changing the length and/or the density a small amount so that the parts do not resonate at the frequency of the motor.
31. Energy is supplied to the glass by the work done by the force of your finger on the glass. When supplied at the right frequency, standing waves form. The glass resonates and the vibrations produce sound.
33. For the equation y(x, t) = 4.00 cm sin(3 m −1 x)cos(4 s −1 t), there is a node because when x = 0.00 m , sin(3 m −1 (0.00 m)) = 0.00, so y(0.00 m, t) = 0.00 m for all time. For the equation y(x, t) = 4.00 cm sin(3 m −1 x + \(\frac{\pi}{2}\))cos(4 s −1 t), there is an antinode because when x = 0.00 m, sin(3 m −1 (0.00 m) + \(\frac{\pi}{2}\)) = + 1.00, so y(0.00 m, t) oscillates between +A and −A as the cosine term oscillates between +1 and -1.
 a. The P-waves outrun the S-waves by a speed of v = 3.20 km/s; therefore, \(\Delta\)d = 0.320 km.
b. Since the uncertainty in the distance is less than a kilometer, our answer to part (a) does not seem to limit the detection of nuclear bomb detonations. However, if the velocities are uncertain, then the uncertainty in the distance would increase and could then make it difficult to identify the source of the seismic waves.
45. v = 1900 m/s, \(\Delta\)t = 1.05 \(\mu\)s
47. y(x, t) = −0.037 cm
49. The pulse will move \(\Delta\)x = 6.00 m.
51. a.  T = 6.98 s
53. A = 0.30 m, \(\lambda\) = 4.50 m, v = 18.00 m/s, f = 4.00 Hz, T = 0.25 s
55. y(x, t) = 0.23 m sin(3.49 m −1 x − 0.63 s −1 t)
57. They have the same angular frequency, frequency, and period. They are traveling in opposite directions and y 2 (x, t) has twice the wavelength as y 1 (x, t) and is moving at half the wave speed.
59. Each particle of the medium moves a distance of 4A each period. The period can be found by dividing the velocity by the wavelength: t = 10.42 s
61. a.  a. A = 0.0125 cm
b. F T = 0.96 N
75. v = 74.54 m/s, \(P_{\lambda}\) = 91.85 W
77. I = 20.0 W/m 2
b. I = \(\frac{P}{A}\), A = 10.0 m2 A = 4\(\pi\)r 2 , r = 0.892 m
79. I = 650 W/m 2
81. \[P \propto E \propto I \propto X^{2} \Rightarrow \frac{P_{2}}{P_{1}} = \left(\dfrac{X_{2}}{X_{1}}\right)^{2}\]
P 2 = 2.50 kW
83. \[I \propto X^{2} \Rightarrow \frac{I_{1}}{I_{2}} = \left(\dfrac{X_{1}}{X_{2}}\right)^{2}\]
I 2 = 3.38 x 10 −5 W/m 2
85. f = 100.00 Hz, A = 1.10 cm
87. a. I 2 = 0.063I 1
b.   a.
b. \(\lambda\) = 2.0 m, A = 4 m
c. \(\lambda_{R}\) = 2.0 m, A R = 6.93 m
101. y R (x, t) = 2A cos \(\left(\dfrac{\phi}{2}\right)\)cos\(\left(kx − \omega t + \dfrac{\phi}{2}\right)\); The result is not surprising because cos(\(\theta\)) = sin(\(\theta + \frac{\pi}{2}\)).
103. \[\begin{split} \lambda_{n} & = \frac{2.00}{n} L, \quad f_{n} = \frac{v}{\lambda_{n}} \\ \lambda_{1} & = 4.00\; m, \quad f_{1} = 12.5\; Hz \\ \lambda_{2} & = 2.00\; m, \quad f_{2} = 25.00\; Hz \\ \lambda_{3} & = 1.33\; m, \quad f_{3} = 37.59\; Hz \end{split}\]
105. v = 158.11 m/s, \(\lambda\)= 4.44 m, f = 35.61 Hz, \(\lambda_{s}\) = 9.63 m
107. y(x, t) = [0.60 cm sin(3 m −1 x)]cos(4 s −1 t)
109. \[\begin{split} \lambda_{100} & = 0.06\; m \\ v & = 56.8\; m/s, \quad f_{n} = n f_{1}, \quad n = 1, 2, 3, 4, 5 \ldots \\ f_{100} & = 947\; Hz \end{split}\]
111.  a. f = 4.74 x 10 14 Hz
b. \(\lambda\) = 422 nm
121. \(\lambda\) = 16.00 m, f = 0.10 Hz, T = 10.00 s, v = 1.6 m/s
123. \(\lambda\) = (v b + v)t b , v = 3.75 m/s, \(\lambda\) = 3.00 m
125. \[\begin{split} \frac{\partial^{2} (y_{1} + y_{2})}{\partial t^{2}} & = -A \omega^{2} \sin (kx - \omega t) - A \omega^{2} \sin (kx - \omega t + \phi) \\ \frac{\partial^{2} (y_{1} + y_{2})}{\partial x^{2}} & = -Ak^{2} \sin (kx - \omega t) - Ak^{2} \sin (kx - \omega t + \phi) \\ \frac{\partial^{2} y(x,t)}{\partial x^{2}} & = \frac{1}{v^{2}} \frac{\partial^{2} y(x,t)}{\partial t^{2}} \\ -A \omega^{2} \sin (kx - \omega t) - A \omega^{2} \sin (kx - \omega t + \phi) & = \left(\dfrac{1}{v^{2}}\right) (-Ak^{2} \sin (kx - \omega t) - Ak^{2} \sin (kx - \omega t + \phi)) \\ v & = \frac{\omega}{k} \end{split}\]
127. y(x, t) = 0.40 m sin(0.015 m −1 x + 1.5 s −1 t)
129. v = 223.61 m/s, k = 1.57 m −1 , \(\omega\) = 142.43 s −1
131. P = \(\frac{1}{2}\)A 2 (2\(\pi\)f) 2 \(\sqrt{\mu F_{T}}\), \(\mu\) = 2.00 x 10 −4 kg/m
133. P = \(\frac{1}{2} \mu\)A 2 \(\omega^{2} \frac{\lambda}{T}\), \(\mu\) = 0.0018 kg/m
135. a. A R = 2A cos\(\left(\dfrac{\phi}{2}\right)\), cos\(\left(\dfrac{\phi}{2}\right)\) = 1, \(\phi\) = 0, 2\(\pi\), 4\(\pi\), ...
b.  a. \[\begin{split} &F_{T} - 20.00\; kg(9.80\; m/s^{2}) \cos 45^{o} & = 0 \\ &m(9.80\; m/s^{2}) - F_{T} = 0 \\ &m = 14.14\; kg \end{split}\]
b. F T = 138.57 N, v = 74.45 m/s
141. F T = 2 N, v = 6.73 m/s
143. a. f n = \(\frac{nv}{2L}\), v = \(\frac{2L f_{n + 1}}{n + 1}\), \(\frac{n + 1}{n} = \frac{2L f_{n + 1}}{2L f_{n}}\), 1 + \(\frac{1}{n}\) = 1.2, n = 5; \(\lambda_{n}\) = \(\frac{2}{n}\)L, \(\lambda_{5}\) = 1.6 m, \(\lambda_{6}\) = 1.33 m
b. F T = 245.76 N
Challenge Problems
145. a. Moves in the negative x direction at a propagation speed of v = 2.00 m/s.
b. \(\Delta\)x = −6.00 m
c.
147. \[\begin{split} \sin(kx - \omega t) & = \sin \left(kx + \dfrac{\phi}{2}\right) \cos \left(\omega t + \dfrac{\phi}{2}\right) - \cos \left(kx + \dfrac{\phi}{2}\right) \sin \left(\omega t + \dfrac{\phi}{2}\right) \\ \sin(kx - \omega t + \phi) & = \sin \left(kx + \dfrac{\phi}{2}\right) \cos \left(\omega t + \dfrac{\phi}{2}\right) + \cos \left(kx + \dfrac{\phi}{2}\right) \sin \left(\omega t + \dfrac{\phi}{2}\right) \\ \sin(kx - \omega t) + \sin(kx - \omega t + \phi) & = 2 \sin \left(kx + \dfrac{\phi}{2}\right) \cos \left(\omega t + \frac{\phi}{2}\right) \\ y_{R} & = 2A \sin \left(kx + \frac{\phi}{2}\right) \cos \left(\omega t + \dfrac{\phi}{2}\right) \end{split}\]
149. sin\(\left(kx + \dfrac{\phi}{2}\right)\) = 0, kx + \(\frac{\phi}{2}\) = 0, \(\pi\), 2\(\pi\), 1.26 m −1 x + \(\frac{\pi}{20}\) = \(\pi\), 2\(\pi\), 3\(\pi\); x = 2.37 m, 4.86 m, 7.35 m
Contributors and Attributions
Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a
Creative Commons Attribution License (by 4.0)
.